Riemann Series Theorem

The Riemann Series Theorem states that if you have a conditionally convergent infinite series—meaning its sum converges only when the terms are added in a specific order—you can rearrange its terms to make it sum to any desired number, or even make it diverge. In other words, the order of addition affects the total sum. This highlights that such series are sensitive to rearrangement, unlike absolutely convergent series, which always sum to the same value regardless of order. It underscores the importance of term arrangement in infinite series and the subtlety of convergence.